{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:A7R7SHX7ILLSUAKEQNTPXDA7AY","short_pith_number":"pith:A7R7SHX7","canonical_record":{"source":{"id":"1409.3875","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-12T21:56:39Z","cross_cats_sorted":[],"title_canon_sha256":"ab88efbb8fe24eac215c5bf81ee1dfc9373d071906751e59953bd2c92bf3cae8","abstract_canon_sha256":"38f5d1de37c0e72481906441ad9a3e79f29d86f2573fc7653281ab7a57bd2f0c"},"schema_version":"1.0"},"canonical_sha256":"07e3f91eff42d72a01448366fb8c1f062a2a1311df8b6f46674195fed3af6744","source":{"kind":"arxiv","id":"1409.3875","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.3875","created_at":"2026-05-18T02:39:18Z"},{"alias_kind":"arxiv_version","alias_value":"1409.3875v2","created_at":"2026-05-18T02:39:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3875","created_at":"2026-05-18T02:39:18Z"},{"alias_kind":"pith_short_12","alias_value":"A7R7SHX7ILLS","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"A7R7SHX7ILLSUAKE","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"A7R7SHX7","created_at":"2026-05-18T12:28:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:A7R7SHX7ILLSUAKEQNTPXDA7AY","target":"record","payload":{"canonical_record":{"source":{"id":"1409.3875","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-12T21:56:39Z","cross_cats_sorted":[],"title_canon_sha256":"ab88efbb8fe24eac215c5bf81ee1dfc9373d071906751e59953bd2c92bf3cae8","abstract_canon_sha256":"38f5d1de37c0e72481906441ad9a3e79f29d86f2573fc7653281ab7a57bd2f0c"},"schema_version":"1.0"},"canonical_sha256":"07e3f91eff42d72a01448366fb8c1f062a2a1311df8b6f46674195fed3af6744","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:18.793033Z","signature_b64":"1cLqWy8rF38Wc+fV/3IO0Sgm3vhHT2rHojHgJW2kDfP8JlykCXOCFdPosow16LuJZf7tSPOdEhxMG9emX//eCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07e3f91eff42d72a01448366fb8c1f062a2a1311df8b6f46674195fed3af6744","last_reissued_at":"2026-05-18T02:39:18.792601Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:18.792601Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.3875","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:39:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ARTjj6XWkLyx6hFxvXEsaY0ppA8YuJgvut5VMe0vW8RenMieaG9szsQzRG15kkQ9h+VeNHMMYdKCXaSzYAoEBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T23:19:25.156813Z"},"content_sha256":"d391b14a240183b58ad282dcbbc9bc38dcddd85948e40ee98cf0bc8baba29c9c","schema_version":"1.0","event_id":"sha256:d391b14a240183b58ad282dcbbc9bc38dcddd85948e40ee98cf0bc8baba29c9c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:A7R7SHX7ILLSUAKEQNTPXDA7AY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$L^{p}$ estimates for the bilinear Hilbert transform for $1/2<p\\leq2/3$: A counterexample and generalizations to non-smooth symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guozhen Lu, Wei Dai","submitted_at":"2014-09-12T21:56:39Z","abstract_excerpt":"M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\\times L^{p_2}\\rightarrow L^{p}$ boundedly when $\\frac{1}{p_1}+\\frac{1}{p_2}=\\frac{1}{p}$ with $1<p_{1}, \\, p_{2}\\leq\\infty$ and $\\frac{2}{3}<p<\\infty$. Whether the $L^p$ estimates hold in the range $p\\in (1/2,2/3]$ has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not map $\\mathcal{F}L^{p'_{1}}\\times L^{p_{2}}\\rightarrow L^{p}$ for $p_1<2$ and $L^{p_{1}}\\times \\mathcal{F}L^{p'_{2}}\\rightarrow L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3875","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:39:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LSTHyLQkX30TMbZft9hNaX5QG7iC8+LbhgsHWsuUTJMNJBbOivbG3YMZo6i/TWnMUfnDoGznGeXvZS22GzKhDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T23:19:25.157157Z"},"content_sha256":"b94f7e1e4b80042fe5a291cc6cb7f41bf5fc6b1a9e6d7fe0ac2eec8360a3d9c7","schema_version":"1.0","event_id":"sha256:b94f7e1e4b80042fe5a291cc6cb7f41bf5fc6b1a9e6d7fe0ac2eec8360a3d9c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY/bundle.json","state_url":"https://pith.science/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T23:19:25Z","links":{"resolver":"https://pith.science/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY","bundle":"https://pith.science/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY/bundle.json","state":"https://pith.science/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A7R7SHX7ILLSUAKEQNTPXDA7AY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:A7R7SHX7ILLSUAKEQNTPXDA7AY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"38f5d1de37c0e72481906441ad9a3e79f29d86f2573fc7653281ab7a57bd2f0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-12T21:56:39Z","title_canon_sha256":"ab88efbb8fe24eac215c5bf81ee1dfc9373d071906751e59953bd2c92bf3cae8"},"schema_version":"1.0","source":{"id":"1409.3875","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.3875","created_at":"2026-05-18T02:39:18Z"},{"alias_kind":"arxiv_version","alias_value":"1409.3875v2","created_at":"2026-05-18T02:39:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3875","created_at":"2026-05-18T02:39:18Z"},{"alias_kind":"pith_short_12","alias_value":"A7R7SHX7ILLS","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"A7R7SHX7ILLSUAKE","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"A7R7SHX7","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:b94f7e1e4b80042fe5a291cc6cb7f41bf5fc6b1a9e6d7fe0ac2eec8360a3d9c7","target":"graph","created_at":"2026-05-18T02:39:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\\times L^{p_2}\\rightarrow L^{p}$ boundedly when $\\frac{1}{p_1}+\\frac{1}{p_2}=\\frac{1}{p}$ with $1<p_{1}, \\, p_{2}\\leq\\infty$ and $\\frac{2}{3}<p<\\infty$. Whether the $L^p$ estimates hold in the range $p\\in (1/2,2/3]$ has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not map $\\mathcal{F}L^{p'_{1}}\\times L^{p_{2}}\\rightarrow L^{p}$ for $p_1<2$ and $L^{p_{1}}\\times \\mathcal{F}L^{p'_{2}}\\rightarrow L","authors_text":"Guozhen Lu, Wei Dai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-12T21:56:39Z","title":"$L^{p}$ estimates for the bilinear Hilbert transform for $1/2<p\\leq2/3$: A counterexample and generalizations to non-smooth symbols"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3875","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d391b14a240183b58ad282dcbbc9bc38dcddd85948e40ee98cf0bc8baba29c9c","target":"record","created_at":"2026-05-18T02:39:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"38f5d1de37c0e72481906441ad9a3e79f29d86f2573fc7653281ab7a57bd2f0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-12T21:56:39Z","title_canon_sha256":"ab88efbb8fe24eac215c5bf81ee1dfc9373d071906751e59953bd2c92bf3cae8"},"schema_version":"1.0","source":{"id":"1409.3875","kind":"arxiv","version":2}},"canonical_sha256":"07e3f91eff42d72a01448366fb8c1f062a2a1311df8b6f46674195fed3af6744","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"07e3f91eff42d72a01448366fb8c1f062a2a1311df8b6f46674195fed3af6744","first_computed_at":"2026-05-18T02:39:18.792601Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:18.792601Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1cLqWy8rF38Wc+fV/3IO0Sgm3vhHT2rHojHgJW2kDfP8JlykCXOCFdPosow16LuJZf7tSPOdEhxMG9emX//eCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:18.793033Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.3875","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d391b14a240183b58ad282dcbbc9bc38dcddd85948e40ee98cf0bc8baba29c9c","sha256:b94f7e1e4b80042fe5a291cc6cb7f41bf5fc6b1a9e6d7fe0ac2eec8360a3d9c7"],"state_sha256":"1ab179db0d199f6ed68dac939c0f5bf8e16595f39e201f58dec46373ba72e3a1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X9hKx7/txwKnWV6TOV5DfWJUzPZqpObc7WbFCFSvNV7b4crg004OpzP8Joc5Y1PiSzNY5cKSqgjouwzKbOS/BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T23:19:25.159094Z","bundle_sha256":"f05745dd8c933d0203de5fc8b9a0063b9ecd066acb0823eafd793d05bcb0e36e"}}